The Gauss-Manin connection for nonabelian cohomology spaces is the isomonodromy flow. We write down explicitly the vector fields of the isomonodromy flow and calculate its induced vector fields on the associated graded space of the nonabelian Hogde filtration. The result turns out to be intimately related to the quadratic part of the Hitchin map.

Let C be an irreducible smooth projective curve, of genus at least two, defined over an algebraically closed field of characteristic zero. For a fixed line bundle L on C, let M C (r; L) be the coarse moduli space of semistable vector bundles E over C of rank r with ∧r E = L. We show that the Brauer group of any desingularization of M C(r; L) is trivial.

Let $X$ be a general proper and smooth curve of genus $2$ (resp. of genus $3$) defined over an algebraically closed field of characteristic $p$. When $3\le p\le 7$, the action of Frobenius on rank $2$ semi-stable vector bundles with trivial determinant is completely determined by its restrictions to the 30 lines (resp. the 126 Kummer surfaces) that are invariant under the action of some order $2$ line bundle over $X$. Those lines (resp. those Kummer surfaces) are closely related to the elliptic curves (resp. the abelian...

This paper deals with rank two connections on the projective line having four simple poles with prescribed local exponents 1/4 and $-1/4$. This Lamé family of connections has been extensively studied in the literature. The differential Galois group of a Lamé connection is never maximal : it is either dihedral (finite or infinite) or reducible. We provide an explicit moduli space of those connections having a free underlying vector bundle and compute the algebraic locus of those reducible connections....

Let C be a smooth projective curve over an algebraically closed field of arbitrary characteristic. Let M r,Lss denote the projective coarse moduli scheme of semistable rank r vector bundles over C with fixed determinant L. We prove Pic(M r,Lss) = ℤ, identify the ample generator, and deduce that M r,Lss is locally factorial. In characteristic zero, this has already been proved by Drézet and Narasimhan. The main point of the present note is to circumvent the usual problems with Geometric Invariant...

Sia $X$ una curva liscia di genere $g\ge 2$ ed $A$, $B$ fasci coerenti su $X$. Sia ${\mu }_{A,B}:H^0\left(X,A\right)\otimes H^0\left(X,B\right)\to H^0\left(X,A\otimes B\right)$ l'applicazione di moltiplicazione. Qui si dimostra che ${\mu }_{A,B}$ ha rango massimo se $A\cong {\omega }_X$ e $B$ è un fibrato stabile generico su $X$. Diamo un'interpretazione geometrica dell'eventuale non-surgettività di ${\mu }_{A,B}$ quando $A,B$ sono fibrati in rette generati da sezioni globali e $\mathrm{deg}\left(A\right)+\mathrm{deg}\left(B\right)\ge 3g-1$. Studiamo anche il caso $\text{dim}\left(\text{Coker}\left({\mu }_{A,B}\right)\right)\ge 2$.

Let C be an elliptic curve and E, F polystable vector bundles on C such that no two among the indecomposable factors of E + F are isomorphic. Here we give a complete classification of such pairs (E,F) such that E is a subbundle of F.